Divide using synthetic division $$ ( 4x^{4}-39x^{3}+30x^{2}-31x+58 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}9&4&-39&30&-31&58\\& & 36& -27& 27& \color{black}{-36} \\ \hline &\color{blue}{4}&\color{blue}{-3}&\color{blue}{3}&\color{blue}{-4}&\color{orangered}{22} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-39x^{3}+30x^{2}-31x+58 }{ x-9 } = \color{blue}{4x^{3}-3x^{2}+3x-4} ~+~ \frac{ \color{red}{ 22 } }{ x-9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&4&-39&30&-31&58\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}9&\color{orangered}{ 4 }&-39&30&-31&58\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 4 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&4&-39&30&-31&58\\& & \color{blue}{36} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -39 } + \color{orangered}{ 36 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}9&4&\color{orangered}{ -39 }&30&-31&58\\& & \color{orangered}{36} & & & \\ \hline &4&\color{orangered}{-3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&4&-39&30&-31&58\\& & 36& \color{blue}{-27} & & \\ \hline &4&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}9&4&-39&\color{orangered}{ 30 }&-31&58\\& & 36& \color{orangered}{-27} & & \\ \hline &4&-3&\color{orangered}{3}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 3 } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&4&-39&30&-31&58\\& & 36& -27& \color{blue}{27} & \\ \hline &4&-3&\color{blue}{3}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -31 } + \color{orangered}{ 27 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}9&4&-39&30&\color{orangered}{ -31 }&58\\& & 36& -27& \color{orangered}{27} & \\ \hline &4&-3&3&\color{orangered}{-4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{9}&4&-39&30&-31&58\\& & 36& -27& 27& \color{blue}{-36} \\ \hline &4&-3&3&\color{blue}{-4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 58 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrr}9&4&-39&30&-31&\color{orangered}{ 58 }\\& & 36& -27& 27& \color{orangered}{-36} \\ \hline &\color{blue}{4}&\color{blue}{-3}&\color{blue}{3}&\color{blue}{-4}&\color{orangered}{22} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-3x^{2}+3x-4 } $ with a remainder of $ \color{red}{ 22 } $.